Profiling
# Install vjsim if you haven't already by running the following commands
# install.packages("devtools")
# devtools::install_github("mladenjovanovic/vjsim")
# Install tidyverse and cowplot packages
# install.packages(c("tidyverse", "cowplot", "DT), dependencies = TRUE)
library(vjsim)
library(tidyverse)
library(cowplot)
library(DT)Before reading this vignette, please read Introduction to vjsim and Simulation in vjsim vignettes by running:
Once we know how vjsim works and how to simulate one jump, we might be interested in checking how jump kinetic parameters change when we add (or remove1) external weight in a form of a barbell or hex-barbell. For the purpose of this vignette, this external load is equal to:
To simulate vertical jump with this external loads, we will used vjsim::vj_profile functions:
profile_data <- vjsim::vj_profile(
external_load = external_load,
# Simulation parameters
mass = 75,
push_off_distance = 0.4,
max_force = 3000,
max_velocity = 4
# Other parameters are default in the `vjsim::fgen_get_output`
# weight = mass * gravity_const,
# gravity_const = 9.81,
# decline_rate = 1.05,
# peak_location = -push_off_distance * 0.15,
# time_to_max_activation = 0.3
)Data frame returned from vjsim::vj_profile is the same as data frame returned from vjsim::vj_simulate with additional two columns: ‘bodyweight’ and ‘external_load’:
datatable(profile_data, rownames = FALSE) %>%
formatRound(columns = 1:ncol(profile_data), digits = 2)Now that we have performed multiple jumps, we can plot various profiles. To save time (and code space), let’s create a function for plotting:
plot_profile <- function(profile_data, x_var, y_var) {
df <- data.frame(
x_var = profile_data[[x_var]],
y_var = profile_data[[y_var]]
)
gg <- ggplot(df, aes(x = x_var, y = y_var)) +
theme_cowplot(8) +
geom_line(color = "blue") +
geom_point(color = "blue") +
labs(x = x_var, y = y_var)
return(gg)
}Load~Height and Load~TOV Profile
The first profile we want to plot is the load~height profile, to see how adding external resistance affects the jump height of this individual athlete:
Rather than using external load on x-axis, we can use total system load, which is equal to bodyweigth plus external load:
As can be seen from both profile, the higher the external load, the lower the jump height. It seems that this profile, or relationship, is curvilinear, although not much. Rather than using height, we can use take-off velocity (TOV):
With this profile, the relationship is more linear. What we want to extract from this profile are the \(Mass_0\) and \(TOV_0\), or the points where this profile line, when extened (or extrapolated), cuts the x- and y-axes. It is easier to explain in picture:
lm_model <- lm(take_off_velocity~mass, profile_data)
model_data <- tibble(
mass = seq(-50, 300),
predicted_tov = predict(lm_model, newdata = data.frame(mass = mass))
)
plot_profile(profile_data, "mass", "take_off_velocity") +
geom_line(data = model_data, aes(y = predicted_tov, x = mass), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
We can do that by using vjsim::get_FV_profile function. vjsim::get_FV_profile utilizes simple linear regression to model the relationship between two variables (mass and TOV in this case) and spits out few summaries:
vjsim::get_FV_profile(
profile_data = profile_data,
force = "mass",
velocity = "take_off_velocity"
)
#> $F0
#> [1] 265.1471
#>
#> $F0_rel
#> [1] 3.535294
#>
#> $V0
#> [1] 3.471573
#>
#> $Pmax
#> [1] 230.1193
#>
#> $Pmax_rel
#> [1] 3.068258
#>
#> $Sfv
#> [1] -76.37664vjsim::get_FV_profile assumes force and velocity variables, and thus can be used to calculate max power (‘Pmax’), but nonetheless it can be used in this scenario (but we need to disregard the Pmax; we will use it in another profile). Here, F0 represents \(Mass_0\), and we can see that, according to this profile and linear regression used, it is equal to 260kg. This means that when total load (i.e. total system mass) is equal to 260kg, TOV will be equal to zero. F0_rel is F0 divided by bodyweight (which is saved in the profile_data object). V0 is the hypothetical take-off velocity when total system load is equal to zero. ‘Sfv’ represents the slope of this regression line. Sfv is equal to -74, which means that for every increase in TOV for 1m/s, total system load needs to be decreased by 74kg.
We can apply this profiling to external load rather than total load:
vjsim::get_FV_profile(
profile_data = profile_data,
force = "external_load",
velocity = "take_off_velocity"
)
#> $F0
#> [1] 190.1471
#>
#> $F0_rel
#> [1] 2.535294
#>
#> $V0
#> [1] 2.489597
#>
#> $Pmax
#> [1] 118.3474
#>
#> $Pmax_rel
#> [1] 1.577965
#>
#> $Sfv
#> [1] -76.37664We can see that now the F0 (or \(Mass_0\)) is equal to 185kg. Now V0 is equal to predicted take-off velocity of the bodyweight jump (since that implies zero external load). It is not exactly equal to TOV from the profile table, since this is modeled value.
For Load~Height profile (acually mass~height model), since it is curvilinear, we can use 2nd degree polynomial regression. Before using vjsim::get_FV_profile, let’s plot the polynomial model and see how it extendes beyond profile data (this is needed to be checked since we will calculate \(Mass_0\) and \(Height_0\)):
lm_model <- lm(height~poly(mass, 2), profile_data)
model_data <- tibble(
mass = seq(-10, 250),
predicted_height = predict(lm_model, newdata = data.frame(mass = mass))
)
plot_profile(profile_data, "mass", "height") +
geom_line(data = model_data, aes(y = predicted_height, x = mass), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
The usual problem with polynomial regressions is exactly this extrapolation beyond the data (which is needed to get \(Mass_0\) and \(Height_0\)). As can be seen from the figure, the 2nd degree polynomial fit never cuts through the x-axis (where Height is equal to 0), and thus \(Mass_0\) cannot be computed. Let’s see whar happens when using 3rd degree polynomial:
lm_model <- lm(height~poly(mass, 3), profile_data)
model_data <- tibble(
mass = seq(-10, 250),
predicted_height = predict(lm_model, newdata = data.frame(mass = mass))
)
plot_profile(profile_data, "mass", "height") +
geom_line(data = model_data, aes(y = predicted_height, x = mass), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
With this 3rd degree polynomial model, \(Mass_0\) and \(Height_0\) can be computed. We can do that with vjsim::get_FV_profile, but we will use ‘poly_deg = 3’ argument to fit 3rd degree polynomial regression model:
vjsim::get_FV_profile(
profile_data = profile_data,
force = "mass",
velocity = "height",
poly_deg = 3
)
#> $F0
#> [1] 225.0424
#>
#> $F0_rel
#> [1] 3.000566
#>
#> $V0
#> [1] 0.7827555
#>
#> $Pmax
#> [1] 44.0383
#>
#> $Pmax_rel
#> [1] 0.5871773
#>
#> $Sfv
#> [1] -287.5003According to this model, when mass is equal to 215kg, jump height is equal to 0m. In practice, we never tend to use load this high, but load that allows for at least 10-15cm (0.1-0.15m) jump height.
To make this simpler, we can use simple linear regression instead to see how this differs:
lm_model <- lm(height~poly(mass, 1), profile_data)
model_data <- tibble(
mass = seq(-10, 250),
predicted_height = predict(lm_model, newdata = data.frame(mass = mass))
)
plot_profile(profile_data, "mass", "height") +
geom_line(data = model_data, aes(y = predicted_height, x = mass), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
And finally calculate \(Mass_0\) and \(Height_0\):
vjsim::get_FV_profile(
profile_data = profile_data,
force = "mass",
velocity = "height"
)
#> $F0
#> [1] 198.4972
#>
#> $F0_rel
#> [1] 2.646629
#>
#> $V0
#> [1] 0.4910784
#>
#> $Pmax
#> [1] 24.36942
#>
#> $Pmax_rel
#> [1] 0.3249256
#>
#> $Sfv
#> [1] -404.2066From these few examples, we can see how profiling depends on the variables used, as well as the models used (i.e. polynomial or linear regression) and thus how can the (extrapolated) parameters differ.
From a simulation perspective, we would be interested to see if \(F_0\) and \(V_0\) can be used to infer underlying (or latent or hidden) Force Generator max_force and max_velocity characteristics. I will explore this in Exploring vignette. For now let’s explore other profiles.
Mean Force~Mean Velocity Profile
Instead of using mass or external load, we can use mean ground reaction force (GRF) over distance2 with mean velocity. Why mean GRF over distance and not over time? Because using mean GRF over distance give correct results for work and thus power calculus, while mean GRF over time does not (but can be used to calculate impulse).
Let’s plot the relationship:
As can be seen from the above figure, from the data simulated using vjsim, it seems that this relationship is not linear, but curvilinear. We do tend to model it with simple linear regression.
lm_model <- lm(mean_velocity~poly(mean_GRF_over_distance, 1), profile_data)
model_data <- tibble(
mean_GRF_over_distance = seq(-50, 3500),
predicted_mv = predict(lm_model, newdata = data.frame(mean_GRF_over_distance = mean_GRF_over_distance))
)
plot_profile(profile_data, "mean_GRF_over_distance", "mean_velocity") +
geom_line(data = model_data, aes(y = predicted_mv, x = mean_GRF_over_distance), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
The extracted \(F_0\) and \(V_0\) are the following:
vjsim::get_FV_profile(
profile_data = profile_data,
force = "mean_GRF_over_distance",
velocity = "mean_velocity"
)
#> $F0
#> [1] 3254.314
#>
#> $F0_rel
#> [1] 43.39085
#>
#> $V0
#> [1] 2.021105
#>
#> $Pmax
#> [1] 1644.327
#>
#> $Pmax_rel
#> [1] 21.92436
#>
#> $Sfv
#> [1] -1610.166If you remember from simulation parameters, Force Generator max force was set to 3000N and max velocity to 4m/s. From this simple example, we can see that manifested F-V relationship doesn’t uncover latent force generator characteristic. This will be further explored in another vignette.
From \(F_0\) and \(V_0\) we can calculate \(P_{max}\), which is equal to \(\frac{F_0 \times V_0}{4}\) (assuming linear relationship). Let’s see what happens when we use 2nd degree polynomial fit:
lm_model <- lm(mean_velocity~poly(mean_GRF_over_distance, 2), profile_data)
model_data <- tibble(
mean_GRF_over_distance = seq(-50, 3500),
predicted_mv = predict(lm_model, newdata = data.frame(mean_GRF_over_distance = mean_GRF_over_distance))
)
plot_profile(profile_data, "mean_GRF_over_distance", "mean_velocity") +
geom_line(data = model_data, aes(y = predicted_mv, x = mean_GRF_over_distance), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
The extracted \(F_0\) and \(V_0\) are now equal to:
vjsim::get_FV_profile(
profile_data = profile_data,
force = "mean_GRF_over_distance",
velocity = "mean_velocity",
poly_deg = 2
)
#> $F0
#> [1] 2716.435
#>
#> $F0_rel
#> [1] 36.21914
#>
#> $V0
#> [1] 1.095512
#>
#> $Pmax
#> [1] 743.9716
#>
#> $Pmax_rel
#> [1] 9.919622
#>
#> $Sfv
#> [1] -2479.604And if we use 3rd degree polynomial fit:
lm_model <- lm(mean_velocity~poly(mean_GRF_over_distance, 3), profile_data)
model_data <- tibble(
mean_GRF_over_distance = seq(-50, 3500),
predicted_mv = predict(lm_model, newdata = data.frame(mean_GRF_over_distance = mean_GRF_over_distance))
)
plot_profile(profile_data, "mean_GRF_over_distance", "mean_velocity") +
geom_line(data = model_data, aes(y = predicted_mv, x = mean_GRF_over_distance), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
The extracted \(F_0\) and \(V_0\) are now equal to:
vjsim::get_FV_profile(
profile_data = profile_data,
force = "mean_GRF_over_distance",
velocity = "mean_velocity",
poly_deg = 3
)
#> $F0
#> [1] 2611.936
#>
#> $F0_rel
#> [1] 34.82582
#>
#> $V0
#> [1] 1.966336
#>
#> $Pmax
#> [1] 1283.986
#>
#> $Pmax_rel
#> [1] 17.11982
#>
#> $Sfv
#> [1] -1328.326As can be seen from the examples above, each model fit provides different estimates for \(F_0\) and \(V_0\) (using vjsim data), but we do tend to represent it with simple linear regression. As can be seen from estimated \(F_0\) and \(V_0\), they do not seem to recover Force Generator max force and max velocity.
Mean Force~Mean Power Profile
As seen from the Mean Force~Mean Velocity Profile, \(P_{max}\) can be calculated using \(F_0\) and \(V_0\). But rather than estimating it, we can actually plot it, since we have that data available from vjsim and in our profilind data frame:
Also, rather than using mean GRF over distance for power profile, we can also use mass and external load:
This is a typical power profile. We can fit 2nd degree polynomial fit to these data points (using mean GRF over distance):
lm_model <- lm(mean_power~poly(mean_GRF_over_distance, 2), profile_data)
model_data <- tibble(
mean_GRF_over_distance = seq(-50, 3500),
predicted_mp = predict(lm_model, newdata = data.frame(mean_GRF_over_distance = mean_GRF_over_distance))
)
plot_profile(profile_data, "mean_GRF_over_distance", "mean_power") +
geom_line(data = model_data, aes(y = predicted_mp, x = mean_GRF_over_distance), alpha = 0.5, linetype = "dashed") +
geom_hline(yintercept = 0, color = "grey", alpha = 0.5) +
geom_vline(xintercept = 0, color = "grey", alpha = 0.5)
From this curve we can extract (or predict) mean \(P_{max}\). For that reason there is vjsim::get_power_profile function:
vjsim::get_power_profile(
profile_data = profile_data,
x_var = "mean_GRF_over_distance",
power = "mean_power"
)
#> $Pmax
#> [1] 1676.722
#>
#> $Pmax_rel
#> [1] 22.3563
#>
#> $Pmax_location
#> [1] 1649.136As can be checked, this \(P_{max}\) and \(P_{max}\) estiamated using \(F_0\) and \(V_0\) (using simple linear regression) is not that different.
According to Maximum Dynamic Output Hypothesis, this peak in mean power (i.e. \(P_{max}\)) happens at one’s own bodyweight(Jaric and Markovic 2009):
Specifically, here we show that the optimal load for the power and momentum production in vertical jumping in habitually active individuals (but not in strength/power-trained athletes) could be the subject’s own body.
But as can be seen, this is not the case in vjsim simulation. I personally see Power as instrumentalist, not realist. In other words, it is just a mathematics not some magical latent property of the Force Generator.
Using Peak rather than Mean values
For far we have used mean force and mean velocity, as well as mean power. We can instead use peak values instead. Here is a profile between peak GRF and peak velocity3:
We can also extract \(F_0\) and \(V_0\) from this relationship (using simple linear regression model):
vjsim::get_FV_profile(
profile_data = profile_data,
force = "peak_GRF",
velocity = "peak_velocity"
)
#> $F0
#> [1] 3113.143
#>
#> $F0_rel
#> [1] 41.50857
#>
#> $V0
#> [1] 6.457276
#>
#> $Pmax
#> [1] 5025.605
#>
#> $Pmax_rel
#> [1] 67.00807
#>
#> $Sfv
#> [1] -482.1139Let’s check the peak power profile:
And if we want to extract the peak value (i.e. peak value of the peak power):
vjsim::get_power_profile(
profile_data = profile_data,
x_var = "peak_GRF",
power = "peak_power"
)
#> $Pmax
#> [1] 3027.243
#>
#> $Pmax_rel
#> [1] 40.36324
#>
#> $Pmax_location
#> [1] 2074.24There are numerous profiles that could be created in vjsim (and with real data). Function vjsim::get_all_profiles runs the most common profiles and returns their results (in a list and a data frame):
